To prove illegal collusion, they have to prove that the Donald Trump campaign or Trump himself made an agreement for someone to engage in an illegal act. Collusion is not in and of itself illegal, the actual crime would be something else.
It is not illegal to hear information that Russia might want to share, even if the information were obtained illegally. It would be illegal to ask Russia to do something illegal to get information.
It is normal for foreign dignitaries to meet with presidential candidates during the campaign to advocate for positions favorable to their countries. This is especially so since many presidential candidates (not Trump or Hillary Clinton in 2016, but Bernie Sanders and Barack Obama and Clinton in 2008) already are in the government when they run.
It is known that Donald Trump Jr. met with these officials in order to get dirt on Hillary. It has recently also become known that George Papadopoulos also met with Russian agents in order to get dirt on Hillary. In both these cases, the particular knowledge that Russia had on Hillary would, obviously, had to have been obtained illicitly by them (such as her private e-mails and what not).
But Russia didn't actually give them any information. Instead, Russia asked them for things (like sanctions relief). To prove collusion, they would have to prove that Don, Jr. or Papadopoulos made an agreement with the Russians. Since neither returned with dirt on Clinton, the meeting was not obviously productive.
An example of what they hope they could find. Rather than using Donald Trump, I'll talk about Bernie Sanders. The same basic argument would apply to Trump, while some of the details may differ.
We’ve also learned that certain pages called for followers to vote for Jill Stein and Bernie Sanders, as opposed to Clinton — although those posts, especially as pertaining to Sanders, haven’t yet been revealed publicly.
This shows that the Russians supported Sanders over Clinton. In fact, Russia hosted Facebook pages and bought ads supporting Sanders over Hillary Clinton both during the primary and after that. (They also supported both Trump and Stein in the general election.)
If Sanders had shared Democratic National Committee information on voters (to which he had access as a nominally Democratic candidate) with the Russians so that they could then use it to target political messaging, that would be collusion. So they would have to prove that Sanders approved the transfer of information to Russia. It would not generally be enough to prove that an employee of his had engaged in the transfer. They would have to prove that Sanders approved of that action, at least implicitly.
When they talk about colluding with the Russians to interfere with the elections, that's what they mean. That a campaign coordinated with the Russians, where the Russians provided advertising and the campaign took advantage of it. Or perhaps the Russians obtained information about voters (e.g. by hacking the state election system) that the campaign then used. They would like to argue that the Russians changed the election results by hacking, but Stein's recounts prevent that claim.
They tend to avoid the idea that the Russians could have placed votes illegally. This is because most of those charging Russian interference are Democrats, and it is the Democratic position that fraudulent votes are rare if not non-existent. This is important to them because Democrat voters have traditionally been more likely to find requests for identification difficult to fill. I.e. Democrats are urban and less likely to have driver's licenses. So a claim of Russians changing the voting results by fraudulent voting might hurt Trump but would help Republican positions overall.
This is of course very difficult to prove. It's not enough to prove that the Russians supported Sanders. That's illegal for the Russians, but it's not illegal for Sanders to receive illegal support. It would be illegal for him to have conspired to get that illegal support from the Russians.
It's not enough to prove meetings, as they did with Trump (if there were meetings with Sanders and Russians, they haven't been published widely). It is not illegal to meet with foreign nationals. For example, Sanders did meet with the Pope, who is currently a resident of the Vatican and is an Argentine. It is illegal to accept campaign donations from them, but there is no proof nor even rumor that the Trump campaign received such donations. They would have to prove that the meetings led to something.
It's quite difficult to prove agreement. Note the politicians who have skated recently, like Sheldon Silver. They proved that he received money and that he provided special treatment but failed to prove that the special treatment was illegally special.
Robert Mueller needs to prove not just a legal meeting, not just a legal political position, not just illegal behavior by the Russians. He needs to prove that the illegal behavior by the Russians was the result of the meeting (or a different meeting, which he'd also have to prove), not because, say, they hate Clinton.
Another way of saying this is that politically, Democrats can accuse Trump of colluding with the Russians based on a meeting with a Russian citizen who knew people in the Russian government. That's just how politics works. The meeting happened. Legally though, proof of that meeting is just a small step in proving a conspiracy (actual legal term) or other crime.
They would have to prove that Don, Jr.'s characterization of the meeting was false. That instead of the Russian lawyer lying about having proof that Clinton was taking money from the Russians when she just wanted to talk about how the Magnitsky Act was impacting adoptions, there was an agreement made whereby someone did something illegal at the behest of Don, Jr. or someone else as a result of a specific agreement.
It's not enough to assert that the Russians would have had to obtained whatever information illicitly. They didn't. If they were laundering funds through intermediaries to the Clinton campaign, the lawyer could have found out about it quite properly. The lawyer could have been considered the representative of a whistleblower in that case.